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In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
Calculus of variations is concerned with variations of functionals, which are small changes in the functional's value due to small changes in the function that is its argument. The first variation [l] is defined as the linear part of the change in the functional, and the second variation [m] is defined as the quadratic part. [26]
Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations: for =, …,. where and are known vectors, denotes equality up to an unknown scalar multiplication, and is a matrix (or linear transformation) which contains the unknowns to be solved.
It is also called the constant of variation or constant of proportionality. Given such a constant k , the proportionality relation ∝ with proportionality constant k between two sets A and B is the equivalence relation defined by { ( a , b ) ∈ A × B : a = k b } . {\displaystyle \{(a,b)\in A\times B:a=kb\}.}
For example, the problem of determining the shape of a hanging chain suspended at both ends—a catenary—can be solved using variational calculus, and in this case, the variational principle is the following: The solution is a function that minimizes the gravitational potential energy of the chain.
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum ( functional derivative equal zero) appears in a weak formulation (variational form) integrated with an arbitrary function δf .
The total derivative is a linear combination of linear functionals and hence is itself a linear functional. The evaluation d f a ( h ) {\displaystyle df_{a}(h)} measures how much f {\displaystyle f} points in the direction determined by h {\displaystyle h} at a {\displaystyle a} , and this direction is the gradient .
Even if linear, it may fail to depend continuously on if and are infinite dimensional (i.e. in the case that (;) is an unbounded linear operator). Furthermore, for Gateaux differentials that are linear and continuous in ψ , {\displaystyle \psi ,} there are several inequivalent ways to formulate their continuous differentiability .